Functoriality in Morse theory on closed manifolds

TL;DR

This paper develops a functorial framework for Morse theory on closed manifolds, establishing chain maps induced by smooth maps and exploring conditions for compositionality, with applications to cohomology and CW structures.

Contribution

It introduces a functorial approach to Morse theory, providing conditions for chain map composition and linking Morse complexes to CW complexes for homology computation.

Findings

Chain maps between Morse complexes are constructed for smooth maps.

Conditions are identified under which composition of chain maps is preserved.

A new proof shows Morse complexes compute manifold homology via CW structures.

Abstract

We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We give sufficient conditions under which composition is preserved. As an application we provide a new proof that the cup product as defined in Morse theory on the chain level agrees with the cup product in singular cohomology. In appendices we present a proof (due to Paul Biran) that the unstable manifolds of a Morse-Smale system are the open cells of a CW structure on the underlying manifold, and also we show that the Morse complex of the triple is canonically isomorphic to the cellular complex of the CW structure. This gives a new proof that the Morse complex is actually a complex and that it computes the homology of the manifold.